When a weight-lifter lifts a weight equal to his upper body weight his trunk makes an angle of 40 degrees, with the vertical, as shown in the diagram below. In this position the erector spinae muscles exert a large force on the sacrum at an angle of 40 degree to the vertical with a moment arm of 50.0 mm about the mid-point between the fourth and fifth lumbar vertebrae. Assuming that the weight-lifter is quasi-static.
(ii) Calculate the direction and magnitude of the joint force at the mid-point between the fourth and fifth lumbar vertebrae by considering the forces and moments acting on the lower body.
To calculate the direction and magnitude of the joint force at the midpoint between the fourth and fifth lumbar vertebrae, we need to consider the forces and moments acting on the lower body.
Let's break down the problem step by step:
Step 1: Identify the forces acting on the lower body.
- The weight of the upper body and the weight lifted are acting downward.
- The joint force at the midpoint between the fourth and fifth lumbar vertebrae is acting upward.
Step 2: Calculate the magnitude of the forces.
- The weight of the upper body is equal to the weight lifted, which is the weight-lifter's upper body weight.
- The weight-lifter's upper body weight can be calculated by multiplying his body weight by the acceleration due to gravity (9.8 m/s^2).
- Let's call the weight-lifter's upper body weight "W" for simplicity.
Step 3: Calculate the moment due to the weight-lifter's upper body weight.
- The moment is calculated by multiplying the magnitude of the force (weight) by the moment arm.
- The moment arm is given as 50.0 mm (millimeters) or 0.05 meters.
- Let's call the moment due to the weight-lifter's upper body weight "M1" for simplicity.
Step 4: Calculate the moment due to the weight lifted.
- The moment is calculated by multiplying the magnitude of the force (weight lifted) by the moment arm.
- The moment arm is the same as the one given earlier and is 50.0 mm or 0.05 meters.
- Let's call the moment due to the weight lifted "M2" for simplicity.
Step 5: Equilibrium condition.
- In a quasi-static equilibrium, the sum of the moments about any point is zero.
- Since we are interested in the joint force at the midpoint between the fourth and fifth lumbar vertebrae, we will use that as our point of analysis.
Step 6: Calculate the joint force at the midpoint between the fourth and fifth lumbar vertebrae.
- Since the sum of the moments about the midpoint is zero, we can set up an equation.
- The equation is M1 + M2 = 0, since the moments are opposite in direction and cancel each other out.
- The joint force is equal in magnitude but opposite in direction to the resultant of the weight of the upper body and the weight lifted.
By following these steps, you can calculate the direction and magnitude of the joint force at the midpoint between the fourth and fifth lumbar vertebrae.